The Nature of Partition Bijections I. Involutions

We analyze involutions which prove several partition identities and describe them in a uniform fashion as projections of “natural” partition involutions along certain bijections. The involutions include those due to Franklin, Sylvester, Andrews, as well as few others. A new involution is constructed for an identity of Ramanujan, and analyzed in the same fashion. Introduction Combinatorial methods in Partition Theory have been developing ever since Sylvester’s magnum opus [S], where a large body of work by Sylvester and his students were presented. Arguably, Franklin’s involution published a year earlier [F] played the most important role in convincing field’s practitioners in the importance of the approach. Franklin’s involution construction was as beautiful and simple as it was mysterious. Undoubtedly, generations of researchers in the field scratched their head trying to explain its origin. This difficulty led to a general understanding that finding similar constructions in other cases is more of an art than a science, and requires a considerable degree of ingenuity. Still, the quest for a beautiful proof led to a successful emulation of the method in a few notable cases. Just a year after Franklin’s involution, Sylvester found an involutive proof of Jacobi triple product identity [S,P1]. Years later, Schur found an involutive proof of an equivalent version of Rogers–Ramanujan identities [Sc,P1]. In modern times, Andrews found an involutive proof of Gauss identity [A1,P1]. Knuth and Paterson found involutive proofs of extensions of Euler’s and Jacobi identities based on a careful analysis of Franklin’s involution [KP]. Most recently, Chen, Hou and Lascoux found a combinatorial proof of another Gauss identity by using a geometric argument of a similar type [CHL]. While all these involutions are in the same spirit, until now there seem to be no natural order among them. The aim of this paper is to bring such an order. In